The power of one conjecture for majority dynamics on random graphs

Let $\rho(k,n)$ be the probability that Red wins when Red's camp initially has size $n/2+k$, with edge probability $p=.5$. The power of one conjecture. There is a constant $b4>0$ such that

$$\rho(1,n)\geq 1/2+b4$$

for all sufficiently large $n$. This conjecture asserts that a single initial-vote advantage gives Red a nontrivial asymptotic advantage; the paper reports numerical evidence but does not establish the claim.